In this paper, we study the metric dimension problem in maximal outerplanar graphs. Concretely, if \(\beta (G)\) denotes of a graph G order n, prove that \(2\le \beta (G) \le \lceil \frac{2n}{5}\rceil \) and bounds are tight. We also provide linear algorithms to decide whether is 2 build resolving set S size \(\lceil for G. Moreover, characterize all graphs with 2.

Let G be a graph with vertex set V(G), and let d(x,y) denote the length of shortest path between nodes x y in G. For positive integer k for distinct x,y∈V(G), dk(x,y)=min{d(x,y),k+1} Rk{x,y}={z∈V(G):dk(x,z)≠dk(y,z)}. A subset S⊆V(G) is k-truncated resolving if |S∩Rk{x,y}|≥1 any pair x,y∈V(G). The metric dimension, dimk(G), minimum cardinality over all sets G, usual dimension recovered when k+1 ...

A family of connected graphs G is said to be a family with constant metric dimension if its metric dimension is finite and does not depend upon the choice of G in G. In this paper, we study the metric dimension of the generalized Petersen graphs P (n, m) for n = 2m + 1 and m ≥ 1 and give partial answer of the question raised in [9]: Is P (n, m) for n ≥ 7 and 3 ≤ m ≤ bn−1 2 c, a family of graphs...

Modifying the Hausdorff-Buseman metric, we obtain a compatible metric with the Fell-Matheron topology on the space of closed subsets of a locally compact Hausdorff second countable space. We also give an alternative expression of this metric, and two more compatible metrics, a metric of summation form and a modified Rockafellar-Wets metric. Through the study of the relation between the original...

We prove that an ultrametric space can be bi-Lipschitz embedded in R if its metric dimension in Assouad’s sense is smaller than n. We also characterize ultrametric spaces up to bi-Lipschitz homeomorphism as dense subspaces of ultrametric inverse limits of certain inverse sequences of discrete spaces.

Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs Gn : F = (Gn)n ≥ 1 depending on n as follows: the order |V (G)| = φ(n) and lim n→∞ φ(n) = ∞. If there exists a constant C > 0 such that dim(Gn) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension, otherwise F has unbou...